Introductory description

Inverse problems play an increasingly important role for modern data oriented applications. Classical examples are medical imaging and tomography where one attempts to reconstruct the internal structure from transmission data.
Using the theory of partial differential equations it is possible to map the unknown internal structure to the observed data. The task of inverting this map is called 'Inverse Problem'.
We will study the mathematical theory that underpins the construction of the forward operator and devise regularisation techniques that will result in well posed inverse problems.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Review of Functional Analysis and PDE theory
• Modelling of simple physical systems, Radon transform
• Loss functions and the direct method
• Regularisation: Tikhonov and Total Variation
• Convergence of solutions for vanishing noise

Learning outcomes

By the end of the module, students should be able to:

• Understand the difference between forward problems and inverse problem
• Derive the loss functional for specific applications
• Apply the direct method to establish the existence of solutions of regularised inverse problems

Interdisciplinary

Inverse problems typically involve physical models. It is critical that students understand the origins of the inverse problems and the implications for the accuracy that the solutions of inverse problems can achieve.

Subject specific skills

Ability to use methods from pde theory
Ability to model simple physical systems
Ability differentiate between noise and bias

Transferable skills

Ability to interpret data. Ability to translate scientific ideas into mathematical language. Ability to think creatively because inverse problems admit many different solutions.